1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Composite Matrix Transformation - Reflection

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Let T1 be the reflection about the line 2x–5y=0 and T2 be the reflection about the line –4x+3y=0 in the euclidean plane.
    (i) The standard matrix of T1 o T2 is: ?

    Thus T1 o T2 is a counterclockwise rotation about the origin by an angle of _ radians?

    (ii) The standard matrix of T2 o T1 is: ?

    Thus T2 o T1 is a counterclockwise rotation about the origin by an angle of _ radians?

    2. Relevant equations

    I think these equations are correct...

    T(v) = A(v)

    A =
    [((2(u_1))^2)), (2(u_1)(u_2)))
    (2(u_1)(u_2)), ((2(u_2))^2))]
    *u being the unit vectors

    Rotation counterclockwise:
    A =
    [cosx -sinx
    sinx cosx]

    S o T is the matrix Transformation with matrix AB

    3. The attempt at a solution

    I thought I understood this, but again, I guess I've understood something incorrectly.

    For the first question, I got the unit vectors to be:
    [(5/sqrt29)], (2/sqrt29)] and [(3/5), (4/5)] for T_1 and T_2 respectively.
    I then got the standard matrix A of T_1 to be:
    [(21/29) (20/29)
    (20/29) (-21/29)]
    and the standard matrix B of T_2 to be:
    [(-7/25) (24/25)
    (24/25) (7/25)]

    I then took AB = the dot product of these matrices to get:
    [(333/6350) (644/6350)
    (-644/6350) (333/6350)]

    I did similar for the second part, but I'll spare all the numbers, since I'm messing something up....

    Further, how would I go about getting the radians? I know the formula for counterclockwise rotation, but wouldn't know how to come up with the radians of such a number...
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Nov 25, 2008 #2


    User Avatar
    Science Advisor

    I don't recognize your formula for the reflection matrix so what I would do is this.
    <3, 4> is a vector in the direction of the line -4x+ 3y= 0 and <-4, 3> is a vector perpendicular to it. The reflection in that line maps <3, 4> into itself and <-4, 3> into its negative, <4, -3> Setting up the two equations
    [tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\left[\begin{array}{c}3 \\ 4\end{array}\right]= \left[\begin{array}{c}3 \\ 4\end{array}\right][/tex]
    [tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\left[\begin{array}{c}-4 \\ 3\end{array}\right]= \left[\begin{array}{c}4 \\ -3\end{array}\right][/tex]
    gives 4 equations for a, b, c, d. I get
    [tex]A= \left[\begin{array}{cc}\frac{-7}{25} & \frac{24}{25} \\ \frac{14}{25} & \frac{7}{25}\end{array}\right][/tex]
    for the first reflection.

    You can do the same for the second reflection and, of course, their composition is the product of the matrices.

    I don't believe
    [(333/6350) (644/6350)
    (-644/6350) (333/6350)]

    is correct because its determinant is not 1, which must be true for a rotation matrix.
  4. Nov 25, 2008 #3

    Thanks again HallsofIvy.

    I used the unit vectors in my formula, and it seems to come out with the same answer; except I tried the technique you gave and I still come up with
    [(-7/25) (24/25)
    (24/25) (7/25)]
    for the second Matrix.

    I guess I'm making some calculation error, as
    [(333/6350) (644/6350)
    (-644/6350) (333/6350)]
    Is the matrix I get from the product AB...

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook